1. **State the problem:** Simplify the expression $\left(5^{-1}\right)\left(5^{-3}\right)$. 2. **Recall the exponent multiplication rule:** When multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^{m+n}$$ 3. **Apply the rule:** $$\left(5^{-1}\right)\left(5^{-3}\right) = 5^{-1 + (-3)} = 5^{-4}$$ 4. **Rewrite the negative exponent:** $$5^{-4} = \frac{1}{5^4}$$ 5. **Calculate $5^4$:** $$5^4 = 5 \times 5 \times 5 \times 5 = 625$$ 6. **Final answer:** $$\left(5^{-1}\right)\left(5^{-3}\right) = \frac{1}{625}$$